Implicit Regularization and Convergence for Weight Normalization

Normalization methods such as batch, weight, instance, and layer normalization are commonly used in modern machine learning. Here, we study the weight normalization (WN) method \cite{salimans2016weight} and a variant called reparametrized projected gradient descent (rPGD) for overparametrized least squares regression and some more general loss functions. WN and rPGD reparametrize the weights with a scale $g$ and a unit vector such that the objective function becomes \emph{non-convex}. We show that this non-convex formulation has beneficial regularization effects compared to gradient descent on the original objective. These methods adaptively regularize the weights and \emph{converge linearly} close to the minimum $\ell_2$ norm solution even for initializations far from zero.